In a way, this talk is a continuation of the Faculty Colloquium talk about Hilbert Schemes. We will use the action of the two-dimensional complex torus on the Hilbert Scheme (induced by the dilating action on the complex plane) to introduce a family of cell decompositions of both the whole Hilbert scheme and of the part of it with support at the origin. The family is  enumerated by the choices of a one-parameter subgroup in the torus, and the cells are enumerated by partitions. These constructions lead to interesting combinatorics.